Probability Asymptotic Theorems and Their Applications

这项研究建议致力于与概率和统计中的一般渐近定理有关的主题，以及它们在各种问题上的应用。概率渐近理论是概率和统计学的核心，在几乎所有科学或社会科学的分支中发挥着中心作用，包括天气预报、气象等。大数定律、中心极限定理和独立随机变量的重对数律等结果形成了现代概率理论。它们已经在许多方向上被推广和推广，其中包括更一般的随机过程和随机测量，它们已经成为渐近统计或大样本理论的基础，大样本理论是评估估计量和统计检验性质的一般框架。在这个框架内，通常假设样本大小n无限增长，并且当样本大小n趋于无穷大时，统计过程的性质在极限内被评估。*本研究建议的第一个焦点涉及我对随机过程的几乎必然和弱收敛的长期研究兴趣，特别是在实值或Banach空间值随机过程的经典极限定理中的大数定律、中心极限定理、重对数律、大偏差概率和中等偏差概率以及精确渐近性。其目的是研究经典极限结果的改进，发展一些新的方法，建立一些新的概率不等式，以证明随机过程的几乎必然和弱收敛。我还打算继续我以前的研究，利用概率论中的现代随机过程技术，找到随机过程几乎必然和弱收敛的条件，并研究这种收敛的统计应用。*这项研究的第二个重点将是研究与高维数据分析问题、分层模型、重采样方法、随机矩阵理论等有关的统计应用中的渐近行为。因此，我建议研究a)自举样本相关矩阵的最大项的渐近定理，b)来自球面系综矩阵乘积的谱半径，c)分层模型，*我所建议的工作的成功完成将是增加我们对随机过程在非常一般和适用的情况下的渐近行为的理解的重要一步，因为渐近定理的结论通常是对可以通过数值方法获得的结论的补充。本提案中将获得的结果将是新颖和重要的，因为它们将扩展、概括和完善文献中的早期工作。所有结果都将在提交给高影响力学术期刊发表的论文中正式公布。

This research proposal is devoted to topics related to general asymptotic theorems in probability and statistics, and their applications to a wide variety of problems. Probability asymptotic theory lies at the heart of probability and statistics, and plays a central stage in almost every branch of science or social science including weather forecasting, psephology, etc. Results such as the law of large numbers, the central limit theorem, and the law of the iterated logarithm for independent random variables have given shape to modern probability theory. They have been extended and generalized in many directions, among others, to more general random processes and random measures, and they have become the basis of asymptotic statistics or large sample theory which is a generic framework for the assessment of properties of estimators and statistical tests. Within this framework, it is typically assumed that the sample size n grows indefinitely, and the properties of statistical procedures are evaluated in the limit as the sample size n tends to infinity.***The first focus of this research proposal relates to my my long-standing research interest in almost sure and weak convergence of random processes, especially in the laws of large numbers, central limit theorems, the law of the iterated logarithm, probabilities of large and moderate deviations, and precise asymptotics in the classical limit theorems for real-valued or Banach space-valued random processes. The goal is to study refinements of the classical limit results, to develop some new methods and to establish some new probability inequalities for proving almost sure and weak convergence of random processes. I also aim to continue my previous research, using modern random process techniques in probability to find conditions under which almost sure and weak convergence holds for random processes, and to investigate statistical applications of such convergence.***A second focus of this research proposal will be on investigating the asymptotic behaviour in statistical applications pertaining to high dimensional data analysis problems, hierarchical models, resampling methods, random matrix theory, etc. Therefore I propose to study the asymptotic theorems for a) largest entries of bootstrapped sample correlation matrices, b) spectral radii for product of matrices from the spherical ensemble, c) hierarchical models, etc.***The successful completion of my proposed work would be an important step in increasing our understanding of the asymptotic behaviour of random processes in very general and applicable situations since the conclusions of an asymptotic theorem often supplement the conclusions which can be obtained by numerical methods. The results to be obtained in this proposal will be novel and significant in that they will extend, generalize, and refine earlier work in the literature. All results will be formalized in papers submitted for publication in high-impact academic journals.